Question

An engineer is designing a pipeline which is supposed to connect two points P and S. The engineer decides to do it in three sections. The first section runs from point P to point Q and costs $54 a mile to lay. The second section runs from point Q to point R and costs $46 a mile. The third one runs from point R to point S and costs $40 a mile. Looking at the diagram below, you see that if you know the lengths marked x and y, then you know the positions of Q and R. Find the values of x and y which minimize the cost of the pipeline.

Answer 1

To minimize the cost of laying a pipeline, one must identify the total cost function in terms of x and y, then use calculus to find the partial derivatives, set them to zero, and solve for x and y.

Step-by-step explanation:

The student's problem involves minimizing the cost of a pipeline that consists of three sections, each with a different cost per mile. This type of problem is typically solved using calculus, particularly optimization. To find the values of x and y that minimize the total cost, you would need to first express the total cost as a function of x and y. However, since no diagram is provided, we cannot construct a specific function without additional information.

Generally, the function would look something like C(x, y) = 54x + 46y + 40z, where C represents the total cost, x and y are the lengths of the first and second sections, and z would be the length of the third section, possibly expressed in terms of x and y if the points P, Q, R, and S form a triangle or some other geometric shape. To minimize C, we usually take the partial derivatives of C with respect to x and y, set them equal to zero, and solve for x and y. This is followed by checking the second derivative or using another method to ensure that the solution indeed gives a minimum cost.